# Teaching Factors and Divisibility

 Would You Rather Listen to the Lesson?

When seventh graders dive into units of factors, fractions, and exponents, one of the earliest lessons focuses on divisibility and factors. This includes the ability to define divisibility rules, factors, as well as GCF and LCM.

To make these early lessons as accessible as possible, math teachers can turn to awesome, diverse teaching strategies. We’ll share a few strategies with you today. Use them and never worry about teaching factors and divisibility again!

## Strategies to Teach Factors and Divisibility

### Divisibility

You can start your lesson on factors and divisibility by explaining what divisibility means. Divisibility rules are a collection of general rules that we use to check if a given number is entirely divisible by another number.

Point out that “divisible by” means that when we divide one number by another, the result is a whole number. That is, a number ‘a’ is divisible by another number ‘b’ if the division a ÷ b is exact (no remainder). For example, 12 ÷ 4 = 3; therefore, 12 is divisible by 4.

Also, 12 is divisible by 3, because we can write the other division 12 ÷ 3 = 4. So, 12 is divisible by both 4 and 3. We can also say 4 and 3 are divisors or factors of 12. Provide a few more examples.

• 15 is divisible by 5, because 15 ÷ 5 = 3 (there is no remainder)
• 15 is not divisible by 7, because 15 ÷ 7 = 2.17 (there is no remainder)
• 0 is divisible by 7, because 0 ÷ 7 = 0 exactly (there is no remainder, and remind students that 0 is a whole number)

#### Divisibility Rules

As mentioned, we use divisibility rules to determine if a number ‘a’ is divisible by another number ‘b’. You can write each rule on the whiteboard or prepare a chart with the rules and hang it on a wall in the classroom. Provide an example as you explain each rule:

1. If the last digit of a number is even, then this number is divisible by 2. For example, in 104 the last digit is even (4), so the number is divisible by 2. 104 ÷ 2 = 52.
2. If the sum of all digits of a given number is divisible by 3, then this number is divisible by 3. If we take 813 as an example, we can see that the sum of all its digits is 12 (8 + 1 + 3 = 12). 12 is a number divisible by 3, so we can infer that 813 is also divisible by 3.
3. If the number that the last two digits in a given number form is divisible by 4, then this number is also divisible by 4. For instance, in 2,152, the number formed by the last two digits is 52. 52 is divisible by 4, i.e. 52 ÷ 4 = 13; therefore, 2,152 is also divisible by 4. 2,152 ÷ 4 = 538.
4. If the last digit in a number is 0 or 5, then this number is divisible by 5. For example, in 495 the last digit is 5, so we can conclude that 495 is divisible by 5. By quickly checking on a calculator, 495 ÷ 5 = 99.
5. If the number is divisible by both 2 and 3, then this number is also divisible by 6. Let’s take 3,672. 3,672 ÷ 2 = 1,863 and 3,672 ÷ 3 = 1,224. Since 3,672 is divisible by 2 and 3, we can say that it’s also divisible by 6.
6. If we take a number, double the last digit, subtract it from the rest of the number (not including the last digit) and the number we end up with by doing so is divisible by 7, the original number is also divisible by 7. For example, if we double the last digit in 623, we’ll get 32 = 6. Then, by subtracting 6 from the rest of the number we’ll get 62 – 6 = 56. 56 is divisible by 7, thus 623 is also divisible by 7.
7. If the number formed by the last three digits in a given number is divisible by 8, then the original number is also divisible by 8. For instance, in 16,512, the number formed by the last three digits is 512. This is a number divisible by 8, i.e. 512 ÷ 8 = 64. From here, we can arrive at the conclusion that 16,512 is also divisible by 8.
8. If in a given number the sum of all digits is divisible by 9, then this number is also divisible by 9. You can take 1,377 as an example. The sum of all digits is 18 (1 + 3 + 7 + 7 = 18). Since we know that 18 is divisible by 9, we can infer that 1,377 is also divisible by 9.
9. If the last digit of a given number is 0, then we know that this number is divisible by 10. For example, in 260, the last digit is 0, therefore 260 is divisible by 10. If you want to quickly check on a calculator 260 ÷ 10 = 26.

### Factors

Once you have provided an overview of divisibility and divisibility rules, you can continue with explaining what factors are. Simply put, a factor is a number that divides exactly into another number, i.e. with no remainder. Provide a few examples of this, such as:

• 3 and 6 are factors of 12, because 12 ÷ 6 = 2 (no remainder) and 12 ÷ 3 = 4 (no remainder)
• 2 and 5 are factors of 10, because 10 ÷ 5 = 2 (no remainder) and 10 ÷ 2 = 5 (no remainder)
• 2 and 4 are factors of 8, because 8 ÷ 2 = 4 (no remainder) and 8 ÷ 4 = 2 (no remainder)

Add that a prime factor is a number that has exactly two different factors, the number itself and 1. For example, the numbers 2, 3, 5, 7, 11, 13, 17, 19, and 23 are considered prime numbers. These are the first few, but there are many others of course.

Finally, explain that a common factor is a number that is a factor of two or more numbers. For instance, if we look at the factors of 12 and 16, we can observe that the factors of 12 are 1, 2, 3, 4, 6, and 12, whereas for 16 they are 1, 2, 4, 8, and 16. Their common factors are 1, 2, and 4.

### Greatest Common Factor & Least Common Multiple

Point out that the greatest common factor (GCF) is the greatest number that divides exactly into two or more numbers or we can say greatest of the common factors of two or more numbers.

Add that the greatest common factor (GCF) is also known as the greatest common divisor (G.C.D) and highest common factor (HCF).

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. The least common multiple is also known as the lowest common denominator (LCD) and lowest common multiple (LCM).

Explain the steps for finding GCF and LCM:

1. First, we’ll make a factor tree for each number. This means that we’ll write all the factors of each number.
2. Then, we’ll determine the prime factors that these numbers have in common.
3. To find the GCF, we’ll look at the factors common for both numbers and simply identify which of these has the greatest value. The answer represents the greatest common factor.
4. To find LCM, we’ll multiply the common factors along with any numbers that are not in common and the answer is LCM. However, we will only multiply each factor the greatest number of times it occurs in either number. In simpler terms, if the number 2 occurs two times in one number, and in the other number it occurs three times, we’ll multiply 2 three times.

#### Example 1 – Finding GCF:

Provide an example of how we apply the above steps in order to find the greatest common factor (GCF). Write two numbers on the whiteboard, such as 10 and 15. Add the factors of 10 (1, 2, 5, 10), as well as of 15 (1, 3, 5, 15).

Point out that now we’ll determine the factors that 10 and 15 have in common. These are 1 and 5. The only thing left to do is identify the greatest one. This is 5. So 5 is the greatest common factor for 10 and 15. We can also say that GCF (10, 15) = 5

#### Example 2 – Finding LCM

After having provided an example of how to identify the greatest common factor, proceed with an example of how to find the least common multiple (LCM). Write two numbers on the whiteboard, such as 18 and 12.

Explain that by using the prime factorization of these two numbers, we’ll construct the smallest number whose prime factorization has all of the ingredients of both these numbers. This number will be the least common multiple.

In simpler terms, after determining the prime factors, we will multiply each factor the greatest number of times it occurs in either number. You can write that the prime factors of 18 = 233, In terms of 12, you can write that 12 = 223.

So LCM will need to have enough prime factors to cover both of these numbers. We see that 18 has one 2 and two 3s, whereas 12 has two 2s and one 3. We’ll multiply the factors the greatest number of times they occur in both 18 and 12, so two 2s and two 3s. In other words:

LCM (18, 12) = 2233

LCM (18, 12) = 36

If you have the technical means in your classroom, you can also enrich your lesson by including videos. For example, use this video to illustrate to students how to check whether random numbers are divisible by 2, 3, 4, 5, 6, 9, and 10 by using the divisibility rules.

In addition, this video contains step-by-step instructions and examples on finding the least common multiple (LCM). Finally, use this video to demonstrate to students how to find the greatest common factor (GCF).

## Activities to Practice Factors and Divisibility

### LCM Game

This is an online game that will help students practice finding the least common multiple of numbers, and thus developing their multiplication and division skills. To implement this game in your classroom, the only thing you’ll need is suitable devices (one per student).

This is an individual game, which makes it a great resource for homeschooling parents as well. Provide instructions for the game to students. The game consists of two rows, there are different pairs of numbers in the upper row and the corresponding LCMs are in the bottom row.

Explain that each pair of numbers should be paired with their correct LCM. Once the child manages to match all numbers with their LCM, the game will continue with another round. Keep playing as long as time allows.

### GCF Jeopardy Game

This online game is in jeopardy game format. In it, students will get to practice their skills at finding the greatest common factor. Make sure each child has a suitable device and explain the game.

Pair students up. Instruct students to choose the 2-player mode from the menu (homeschooling parents can choose one player). Each player chooses their avatar. Then, they take turns answering questions with their avatar. Students click on their avatar before selecting a question.

Depending on the difficulty of the question, they can score up to 400 points. Each question is related to finding the greatest common factor of a set of given numbers. There are multiple-choice answers from which the student must select one.

In the end, students compare their scores. In each pair, the student with the highest score wins the game. You can also introduce symbolic prizes for the winners.

### ‘Am I Divisible By Him?’ Game

This is a game where students will sharpen their ability to quickly check whether a number is divisible by another number with the help of the divisibility rules. To use the game in your classroom, you’ll need to prepare plenty of task cards (around 20 cards per pair).

Divide students into pairs and place the cards in the middle, facing down. Each card contains two numbers, for example, 21; 872; 4. In each of the numbers on the cards, without doing actual division, students have to determine whether the first number is divisible by the second number.

Player one draws the first card and has 1 minute to answer the question on the card they selected. If they answer correctly, they score one point; if they answer incorrectly, they lose two points. Player two repeats the procedure.

Make sure there is a ‘checker’ with the answer sheet in each group. In the end, the student with the greatest score wins the game. A new round begins. The winner becomes the ‘checker’, whereas the student that was a ‘checker’ thus far joins the new round as a player.

Create space for discussion and reflection at the end of the game. How did students know whether a certain number was divisible by another one? How did they know that a certain number wasn’t divisible by another one? Which divisibility rules did they apply to check this?

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